The deformation quantization mapping of Poisson- to associative structures in field theory

Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $\pi$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the Poisson algebra $\boldsymbol{\mathcal{A}}$ of local functionals $\Gamma(\pi)\to\Bbbk$ that take field configurations to numbers. By applying the techniques from geometry of iterated variations, we make well defined the deformation quantization map ${\times}\mapsto{\star}={\times}+\hbar\,\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}+\bar{o}(\hbar)$ that produces a noncommutative $\Bbbk[[\hbar]]$-linear star-product $\star$ in $\boldsymbol{\mathcal{A}}$.Comment: Proc. 50th Sophus Lie Seminar (26-30 September 2016, Bedlewo, Poland), 8 figures, 24 page

The geometry of variations in Batalin-Vilkovisky formalism

This is a paper about geometry of (iterated) variations. We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "$\delta(0)=0$" and "$\log\delta(0)=0$" within BV-approach to quantisation of gauge systems. Remarkably, the geometry of iterated variations does not refer at all to the construction of Dirac's $\delta$-function as a limit of smooth kernels. We illustrate the reasoning by re-deriving - but not just "formally postulating" - the standard properties of BV-Laplacian and Schouten bracket and by verifying their basic inter-relations (e.g., cohomology preservation by gauge symmetries of the quantum master-equation).Comment: XXI International Conference on Integrable Systems and Quantum Symmetries (ISQS21) 11-16 June 2013 at CVUT Prague, Czech Republic; 51 pages (9 figures). - Main Example 2.4 on pp.34-36 retained from arXiv:1302.4388v1, standard proofs in Appendix A amended and quoted from arXiv:1302.4388v1 (joint with S.Ringers). - Solution to Exercise 11.6 from IHES/M/12/13 by the same autho

The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.Comment: Talks given at Mathematics seminar (IHES, 25.11.2016) and Oberseminar (MPIM Bonn, 2.02.2017), 23 figures, 60 page

The Jacobi identity for graded-commutative variational Schouten bracket revisited

This short note contains an explicit proof of the Jacobi identity for variational Schouten bracket in $Z_2$-graded commutative setup. For the reasoning to be rigorous, it refers to the product bundle geometry of iterated variations (see arXiv:1312.1262 [math-ph]); no ad hoc regularizations occur anywhere in this theory.Comment: Addendum to arXiv:1312.1262 [math-ph] (proof of Theorem 4.iii) by the same author; 4+iii pages. - Proc. Int. Workshop SQS'13 `Supersymmetry and Quantum Symmetries' (July 29 -- August 3, 2013; JINR Dubna, Russia

Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?

From the paper "Formality Conjecture" (Ascona 1996): "I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with $4$ vertices $($and $6$ edges$)$. This class gives a remarkable vector field on the space of bi-vector fields on $\mathbb{R}^{d}$. The evolution with respect to the time $t$ is described by the following non-linear partial differential equation: ..., where $\alpha=\sum_{i,j}\alpha_{ij} {\partial}/{\partial x_{i}}\wedge {\partial}/{\partial x_{j}}$ is a bi-vector field on $\mathbb{R}^d$. It follows from general properties of cohomology that $1)$ this evolution preserves the class of $($real-analytic$)$ Poisson structures}, ... In fact, I cheated a little bit. In the formula for the vector field on the space of bivector fields which one get from the tetrahedron graph, an additional term is present. ... It is possible to prove formally that if $\alpha$ is a Poisson bracket, i.e. if $[{\alpha,\alpha}]=0\in T^2(\mathbb{R}^d)$, then the additional term shown above vanishes." By using twelve Poisson structures with high-degree polynomial coefficients as explicit counterexamples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at Poisson bi-vectors. The counterexamples at hand suggest a correction to the formula for the "exotic" flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance $1:6$ for which the flow does preserve the space of Poisson bi-vectors.Comment: Talks given in parallel by A.B. at GADEIS VIII workshop (12--16 June 2016, Larnaca, Cyprus) and A.K. at ISQS'24 conference (13--19 June 2016, CVUT Prague, Czech Republic), 10 pages, 2 figures, 4 table